Linear blocking sets in P G(2, q 4 )

Linear blocking sets in P G(2, q ⁴ )

P. Polito and O. Polverino

Dipartimento di Matematica Via Vivaldi, 81100 Caserta

ITALY [email protected]

Abstract

In this paper, by using the geometric construction of linear blocking sets as projections of canonical subgeometries, we determine all the GF (q)- linear blocking sets of the plane P G(2, q⁴).

1 Introduction

A blocking set B in the projective plane P G(2, q) is a set of points meeting every line. B is called trivial if it contains a line, and it is called minimal if no proper subset of it is a blocking set. We say B is small when its size is less than 3(q + 1)/2 and we call B of R´edei type if there exists a line l such that |B \ l| = q (the line l is called a R´edei line of B).

A family of small minimal blocking sets in P G(2, q^t), called GF (q)-linear blocking sets, was introduced by G. Lunardon in [3] (for a survey on linear blocking sets see [8]). Every small minimal blocking set of R´edei type in P G(2, q), q = pⁿand p
= 2, 3, is a linear blocking set over some non-trivial subfield of GF (q) (see [1] and [3]), and the known examples of small minimal blocking sets not of R´edei type are linear ([9]).

Hence, all the presently known small minimal blocking sets are linear.

In the planes P G(2, q²) and P G(2, q³), the GF (q)-linear blocking sets are com- pletely classified: in P G(2, q²) they are Baer subplanes and in P G(2, q³) they are isomorphic either to the blocking set obtained from the trace function of GF (q³) over GF (q) or to the blocking set obtained from the function x → x^q([10]).

In this paper, we study the GF (q)-linear blocking sets in P G(2, q⁴). Our main result is the following theorem:

Theorem 1.1 Let B be a GF (q)-linear blocking set in P G(2, q⁴). If B is of R´edei type with at least two R´edei lines, then either B is a Baer subplane or B has q⁴+q³+1 points, q + 1 R´edei lines and it is equivalent to the blocking set obtained from the graph of the trace function of GF (q⁴) over GF (q). If B is of R´edei type with a unique R´edei line, then the possible sizes of B are q⁴+ q³+ 1 and q⁴+ q³+ q²+ cq + 1 with c ∈ {−1, 0, 1}. Finally, if B is not of R´edei type, then B has size q⁴+ q³+ q²+ dq + 1 with d ∈ {0, 1}.

Finally, in the last section we prove that there exists at least one example of GF (q)- linear blocking set in P G(2, q⁴) for each possible cardinality.

2 Linear blocking sets in P G(2, q

)

It is possible to define a linear blocking set in three different and equivalent ways. In this paper we will use the geometric construction of such blocking sets following [4].

Let Σ = P G(t, q), t ≥ 3, be a canonical subgeometry of Σ^∗ = P G(t, q^t). Let Λ be a (t − 3)−dimensional subspace of Σ^∗disjoint from Σ, and let π be a plane of Σ^∗ disjoint from Λ. Recall that the projection of Σ from the axis Λ to the plane π is the map pΛ,π,Σfrom Σ to π defined by

pΛ,π,Σ(P ) =< P, Λ > ∩π

for each point P of Σ. The set pΛ,π,Σ(Σ) is a GF (q)−linear blocking set of π = P G(2, q^t). Also, any GF (q)−linear blocking set of P G(2, q^t) can be constructed as a projection of a suitable canonical subgeometry of P G(t, q^t) (see [5]). Note that, since Σ is a canonical subgeometry, there is no hyperplane of Σ^∗ containing Σ and hence the GF (q)-linear blocking sets obtained projecting Σ are non-trivial.

Now, suppose that t = 4 and let Σ = P G(4, q) be a canonical subgeometry of Σ^∗ = P G(4, q⁴). Let σ be the unique semilinear collineation of Σ^∗ which fixes Σ pointwise. Then σ⁴= 1 and the set of fixed points of σ² is a canonical subgeometry Σ^= P G(4, q²) of Σ^∗ containing Σ. Also, if Si is an i-dimensional subspace of Σ^∗, then Si∩ Σ (resp. Si∩ Σ^) is an i-dimensional subspace of Σ (resp. Σ^) if and only if S_i^σ= Si (resp. S_i^σ2 = Si) (see e.g. [4]).

Let π be a plane of Σ^∗and let l be a line disjoint from both π and Σ. Denote by p the projection pl,π,Σof Σ from l to the plane π and by Blthe GF (q)-linear blocking set p(Σ) of the plane π. Notice that if R is a point of Bl, then p⁻¹(R) is an i-dimensional subspace of Σ with i ∈ {0, 1, 2}, and < p⁻¹(R) > is an i-dimensional subspace of Σ^∗ containing l. Similarly, if r is a line of π, then p⁻¹(r ∩ Bl) is an i-dimensional subspace of Σ with i ∈ {0, 1, 2, 3}, and < p⁻¹(r ∩ Bl) > is an i-dimensional subspace of Σ^∗containing l.

Proposition 2.1 Blis of R´edei type if and only if l is contained in a 3-dimensional subspace of Σ^∗fixed by σ.

Proof. Suppose that there exists a 3-dimensional subspace S3 of Σ^∗ containing l and fixed by σ. Then S3∩Σ is a 3-dimensional subspace of Σ projected from l to the line r = π ∩ S3, i.e. p⁻¹(r ∩ Bl) = S3∩ Σ. Let R be a point of Bl\ r. Since p⁻¹(R) is a subspace of Σ disjoint from p⁻¹(r ∩ Bl), p⁻¹(R) is a point of Σ. This implies that

|Bl\ r| = |Σ \ S3| = q⁴, i.e. r is a R´edei line of Bl.

Now, suppose that Blis of R´edei type and let r be a R´edei line of Bl. Since Blis non-trivial, it contains at least q⁴+ q²+ 1 points (see [2]) and|Bl∩ r| ≥ q²+ 1. This implies that p⁻¹(r ∩ Bl) is either a 2 or a 3-dimensional subspace of Σ. In the latter case, < p⁻¹(r ∩ Bl) > is a 3-dimensional subspace of Σ^∗containing l and fixed by σ.

In the former case, the q⁴+ q³points of Σ\ p⁻¹(r ∩ Bl) are projected to the q⁴points of Bl\ r. Hence, there exist at least two points R and T of Bl\ r such that p⁻¹(R) and p⁻¹(T ) contain some line of Σ. Let n and t be lines of Σ^∗such that Σ∩ n is a line of p⁻¹(R) and Σ ∩ t is a line of p⁻¹(T ). Then n ∩ t = ∅, n ∩ l
= ∅ and t ∩ l
= ∅.

Hence S3 =< n, t > is a 3-dimensional subspace of Σ^∗ containing l fixed by σ.

Corollary 2.2 Bl is of R´edei type if and only if dim < l, l^σ, l^σ2, l^σ3 >≤ 3. Also, if Bl is not a Baer subplane, then it has a unique R´edei line if and only if dim <

l, l^σ, l^σ2, l^σ3 >= 3.

Proof. By Proposition 2.1, if Bl is of R´edei type, then l is contained in some 3- dimensional subspace, say S3, of Σ^∗ fixed by σ. This implies that l, l^σ, l^σ2, and l^σ3 are contained in S3 and hence dim < l, l^σ, l^σ2, l^σ3 >≤ 3. Conversely, suppose that dim < l, l^σ, l^σ2, l^σ3 >≤ 3. If dim < l, l^σ, l^σ2, l^σ3 >= 3, then S3 =< l, l^σ, l^σ2, l^σ3 >

is a 3-dimensional subspace of Σ^∗ containing l and fixed by σ, hence Blis of R´edei type. If dim < l, l^σ, l^σ2, l^σ3 >= 2, for any point P ∈ Σ\ < l, l^σ, l^σ2, l^σ3 >, it is easy to check that < P, l, l^σ, l^σ2, l^σ3 > is a 3-dimensional subspace fixed by σ, so Bl is of R´edei type. Finally, suppose that Blis of R´edei type, but it is not a Baer subplane.

In this case|Bl| ≥ q⁴+ q³+ 1 (see [1]) and hence, if r is a R´edei line of Bl, then

|Bl∩ r| ≥ q³+ 1. This implies that p⁻¹(r ∩ Bl) is a 3-dimensional subspace of Σ, hence < p⁻¹(r ∩ Bl) > is a 3-dimensional subspace of Σ^∗ containing l fixed by σ.

Then < l, l^σ, l^σ2, l^σ3 >⊂< p⁻¹(r ∩ Bl) >. Thus r is the unique R´edei line of Blif and only if < l, l^σ, l^σ2, l^σ3 >=< p⁻¹(r ∩ Bl) >, i.e. if and only if dim < l, l^σ, l^σ2, l^σ3 >= 3.

Proposition 2.3 With the previous notation, the following are equivalent:

1) B has maximum size q⁴+ q³+ q²+ q + 1.

2) There is no line of Σ projected from l to a point of π.

3) There is no line of Σ^∗ fixed by σ incident with l.

Proof. Bl has maximum size if and only if the map p is a bijection, hence if and only if p⁻¹(R) is a point of Σ for every point R ∈ Bl (see also [4]). Finally, Bl has not maximum size if and only if there exists a line t of Σ projected from l to a point of Bl. In this case, if t^is the line of Σ^∗ containing t, then t^ is incident with l and

t^σ= t^.

3 Proof of Theorem 1.1

The structure of Bldepends on the position of l with respect to the subgeometries Σ and Σ^. In order to determine the different possibilities for Bl, we have to distinguish between the following cases:

(A) l = l^σ2 ⇐⇒ l intersects Σ^in a line;

(B) l ∩ l^σ2 is a point ⇐⇒ l intersects Σ^in a point;

(C) l ∩ l^σ2=∅ ⇐⇒ l is disjoint from Σ^.

Case A

It is easy to check that in this case l ∩ l^σ=∅, hence S3=< l, l^σ> is a 3-dimensional subspace of Σ^∗. Since S3 is fixed by σ, by Proposition 2.1, Bl is of R´edei type and r = S3∩ π is a R´edei line of Bl. Also, if P ∈ l ∩ Σ^ the line < P, P^σ > is fixed by σ and hence < P, P^σ> ∩Σ is a line of Σ. So < P, P^σ> ∩Σ is projected from l to a point of r ∩ Bl, for every point P ∈ l ∩ Σ^. This implies that the size of Bl∩ r is q²+ 1, and hence|Bl| = q⁴+ q²+ 1, i.e. Blis a Baer subplane of π ([2]).

Case B

Put l ∩ l^σ2 ={P }. Note that the line < P, P^σ > is fixed by σ, hence < P, P^σ >

intersects Σ in a line.

(B1) l ∩ l^σ
= ∅

In this case, we have l^σ∩ l^σ2
= ∅, l^σ2 ∩ l^σ3
= ∅, and l^σ3∩ l
= ∅, i.e. π =<

l, l^σ, l^σ2, l^σ3 > is a plane of Σ^∗fixed by σ. Hence π ∩ Σ is projected from l to a point R of Bl. Also, every 3-dimensional subspace obtained joining π to a point of Σ\ π intersects Σ in a 3-dimensional subspace. Then, through R there pass q + 1 R´edei lines and |Bl| = q⁴+ q³+ 1. This implies that Blis equivalent to the blocking set obtained from the graph of the trace function of GF (q⁴) over GF (q) (see [6]).

(B2) l ∩ l^σ=∅ and dim < l, l^σ, l^σ2, l^σ3 >= 3

Let S3=< l, l^σ, l^σ2, l^σ3 >. By Corollary 2.2 and Proposition 2.1, Blis of R´edei type and S3∩ π is a R´edei line of Bl. Note that both lines < P, P^σ > and l^=< l, l^σ2 > ∩ < l^σ, l^σ3 > are fixed by σ, so they intersect Σ in a line. Also, any line of Σ^∗ fixed by σ and incident with l is incident with l^σ, l^σ2 and l^σ3. This implies that < P, P^σ > and l^ are the unique lines of Σ^∗ fixed by σ and incident with l.

(B21) If < P, P^σ>= l^, then exactly one line of Σ is projected from l to a point of Bl, so Blhas size q⁴+ q³+ q²+ 1.

(B22) If < P, P^σ >
= l^, then exactly two lines of Σ are projected from l to a point of Bl, so Bl has size q⁴+ q³+ q²− q + 1.

Since the blocking sets Bl obtained in Cases B21 and B22 are not Baer sub- planes, S3∩ π is the unique R´edei line of Bl (Corollary 2.2).

(B3) l ∩ l^σ=∅ and dim < l, l^σ, l^σ2, l^σ3 >= 4

Since the subspace joining l, l^σ, l^σ2, and l^σ3 has dimension four, Bl is not of R´edei type (Corollary 2.2). If m is a line fixed by σ and incident with l, then m =< P, P^σ>. Thus, Blhas size q⁴+ q³+ q²+ 1.

Case C

(C1) dim < l, l^σ, l^σ2, l^σ3>= 3

Let S3 =< l, l^σ, l^σ2, l^σ3 >. In this case Bl is of R´edei type and r = S3∩ π a R´edei line of Bl (Corollary 2.2).

(C11) Suppose that l ∩ l^σ
= ∅ and let {P } = l ∩ l^σ. This implies that l =<

P, P^σ3 >. Hence the unique lines intersecting l, l^σ, l^σ2 and l^σ3 are <

P^σ2, P > and < P^σ3, P^σ >. Since such lines are not fixed by σ, there is no line of Σ^∗projected from l to a point of Bl, i.e. Blhas maximum size (Proposition 2.3).

(C12) Suppose that l ∩ l^σ = ∅ and that l, l^σ, l^σ2 and l^σ3 belong to the same regulusR of S3. SinceR is fixed by σ, R ∩ Σ is a regulus of S3∩ Σ. This implies that each transversal line toR ∩ Σ is projected from l to a point of r ∩ Bl. Since the transversal lines toR ∩ Σ number q + 1, the size of r ∩ Blis q³+ 1, and|Bl| = q⁴+ q³+ 1 (see also [6]).

Now, suppose that l ∩ l^σ =∅ and that l, l^σ, l^σ2 and l^σ3 do not belong to the same regulus of S3. Let R be the regulus determined by l, l^σ, and l^σ2 and let R be the opposite regulus to R. A line l^ fixed by σ and incident with l, is incident with l^σ, l^σ2 and l^σ3 and hence it is a transversal line to R, R^σ, R^σ2 andR^σ3, i.e. l^∈ R ∩ R^σ∩ R^σ2∩ R^σ3. Note that two distinct reguli can have at most two transversal lines in common and that the intersection ofR, R^σ, R^σ2 andR^σ3 is fixed by σ.

(C13) R, R^σ,R^σ2 andR^σ3have two transversal lines in common, both fixed by σ. Then Blhas size q⁴+ q³+ q²− q + 1.

(C14) R, R^σ,R^σ2 andR^σ3 have two transversal lines in common, each one not fixed by σ. Then Blhas maximum size.

(C15) R, R^σ, R^σ2 and R^σ3 have a unique transversal line in common. Such transversal is fixed by σ, so Bl has size q⁴+ q³+ q²+ 1.

(C16) R, R^σ, R^σ2 andR^σ3 have no transversal line in common. Then Bl has maximum size.

Since the blocking sets Bl obtained in Cases (C1i), for i = 1, . . . , 6, are not Baer subplanes, r = S3∩ π is the unique R´edei line of Bl(Corollary 2.2).

(C2) dim < l, l^σ, l^σ2, l^σ3>= 4

By Corollary 2.2, Bl is not of R´edei type. Also, l, l^σ, l^σ2 and l^σ3 are pairwise disjoint. Let S3 =< l, l^σ > and let L = S3∩ S3^σ∩ S^σ3² ∩ S3^σ3. Note that L is fixed by σ and, since S3^σ
= S3, dim L ∈ {0, 1, 2}. If dim L = 2, then S3∩ S3^σ = S3^σ2∩ S3^σ3. Hence l^σ and l^σ3 are contained in the plane S3∩ S3^σ, a contradiction. So, 0≤ dim L ≤ 1. If l^is a line fixed by σ and incident with l, then l^is contained in L.

(C21) Suppose that dim L = 1. Then L is the unique line of Σ projected from l to a point of Bl. So|Bl| = q⁴+ q³+ q²+ 1.

(C22) Suppose that dim L = 0. In this case there is no line of Σ projected from l to a point of Bl. Hence Bl has maximum size.

This completes the proof of Theorem 1.1.

4 Examples

In this section we show that all the cases discussed in the proof of Theorem 1.1, but Case (C16), effectively occur.

Let σ be the semilinear collineation of Σ^∗= P G(4, q⁴) defined by σ : (x0, x1, x2, x3, x4) −→ (x^q0, x^q4, x^q1, x^q2, x^q3).

Then the set Σ ={(α, x, x^q, x^q2, x^q3) : α ∈ GF (q), x ∈ GF (q⁴)} of fixed points of σ is a 4-dimensional canonical subgeometry of Σ^∗. Let l be the line with equations

x0= 0, x1= βx3, x2= ax3+ bx4,

where β, a, b ∈ GF (q⁴). The lines l, l^σ, l^σ2and l^σ3 are contained in the 3-dimensional subspace with equation x0 = 0. Hence, if l ∩ Σ = ∅, projecting Σ from l to a plane π disjoint from l, we obtain a GF (q)-linear blocking set of R´edei type of π. By different choices of the coefficients β, a and b we get all the GF (q)-linear blocking sets of R´edei type listed in Theorem 1.1, but Case (C16):

• If β = 1, a = 0, b^q2+1 = 1 and b
= 1, then l ∩ Σ = ∅, l = l^σ2 and hence Case (A) occurs.

• If β = 0 and b^q2+1= 1, then l ∩ Σ = ∅ and l ∩ l^σ2 ={P } with P = (0, 0, b, 0, 1).

In this case, if a = b = −1, since l ∩ l^σ
= ∅, we get Case (B1). If a = 1 and b
= −1, since l ∩ l^σ=∅ and P^σ∈< l, l^σ2>, we get Case (B21). Finally, if b = 1 and a
∈ GF (q²), since l ∩ l^σ=∅ and P^σ
∈< l, l^σ2>, we get Case (B22).

• If β = a = b = 0, then l ∩ Σ = ∅, l ∩ l^σ2 =∅ and l ∩ l^σ
= ∅. So Case (C11) occurs.

• If β = a = 0 and b^q2+1
= 1 with b
= 0, then l∩Σ = ∅, l, l^σand l^σ2are mutually disjoint and determine a regulusR of the quadric with equations

x0= 0, b^q2+q+1x1x2− b^q+1x1x4− x2x3+ bx3x4= 0.

If b^q3+q2+q+1= 1, then l^σ3∈ R (Case (C12)). If b^q3+q2+q+1
= 1 then R, R^σ,R^σ2 andR^σ3 have in common the transversal lines with equations x0= x4= x2= 0 and x0= x3= x1= 0, each one not fixed by σ (Case (C14)).

• If β = b = 0 and a
= 0, then l ∩ Σ = ∅, l, l^σ and l^σ2 are mutually disjoint and determine a regulusR of the quadric with equations

x0= 0, ax²3− a^q2+qx1x2+ a^qx2x4− x2x3− a^q+1x3x4= 0.

By direct calculations it is possible to prove that if either q is even or q is odd and 1 + 4a^q3+q2+q+1is a non square in G(q), then the reguli R, R^σ,R^σ2 and R^σ3 have two transversal lines in common, both fixed by σ (Case (C13)). Also, if q is odd and a^q3+q2+q+1=−1/4, then the reguli R, R^σ,R^σ2 andR^σ3 have a unique transversal line in common (Case (C15)).

In order to obtain the GF (q)-linear blocking sets not of R´edei type consider the following lines:

• Let l be the line of Σ^∗with equations x0= x4, x1= x3, x2= 0. Since l∩Σ = ∅, l ∩ l^σ2 ={P } with P = (0, 1, 0, 1, 0), l ∩ l^σ=∅ and dim < l, l^σ, l^σ2, l^σ3 >= 4, projecting Σ from l to a plane π disjoint from l, we obtain a GF (q)-linear blocking set of π as discussed in Case (B3).

• Let l be the line of Σ^∗ with equations x0 = x4, x1 = 0, x2 = 0 and let S3 =< l, l^σ >. Since l ∩ Σ = ∅, l ∩ l^σ2 = ∅, dim < l, l^σ, l^σ2, l^σ3 >= 4 and dim(S3∩ S3^σ∩ S3^σ2∩ S3^σ3) = 0, projecting Σ from l to a plane disjoint from l, we get Case (C22).

• Finally, let l be the line with equations x1 = ax0, x1 = x2 and x2 = x3

with a
∈ GF (q²) and let S3 =< l, l^σ >. Since l ∩ l^σ2 = ∅, l ∩ l^σ = ∅ and L = S3∩ S3^σ∩ S3^σ2 ∩ S^σ3³ is the line with equations x1 = x2, x2 = x3 and x3= x4, Case (C21) occurs.

We close this section by noting that different positions of the axis of the projection with respect to Σ and Σ^can produce non-equivalent linear blocking sets of the same type and of the same size. Indeed, projecting Σ to the plane π with equations x3= x4= 0 from the line la,bwith equations x0= 0, x1= 0, x2= ax3+ bx4, we get the following GF (q)-linear blocking set of π

Bla,b={(α, x, x^q− ax^q2− bx^q3) : α ∈ GF (q), x ∈ GF (q⁴)}.

As previously noted, if a = b = 0, then Bl0,0 is a GF (q)-linear blocking set of R´edei type of maximum size of Case (C11) and, if a = 0 and b^q3+q2+q+1
= 1, Bl0,bis a GF (q)- linear blocking set of R´edei type of maximum size of Case (C14). It is possible to prove (see [7]) that Bl0,0 and Bl0,b with b^q3+q2+q+1
= 1 are not isomorphic if q > 3.

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(Received 14/3/2001)